Doctoral Theses

Reality Properties of Conjugacy Classes in Algebraic Groups.

4-28-2006

4-28-2007

Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

Degree Name

Doctor of Philosophy

Mathematics

Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

Supervisor

Thakur, Maneesh (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

In this thesis we denote a field by k. We consider fields of characteristic not 2 unless stated otherwise. The notation Â¯k and ks denotes an algebraic closure and separable closure of k respectively. The symbols Q, R, C will denote fields of rational, real, complex numbers respectively. The symbol Z will denote the set of integers. We denote by cd(k) the cohomological dimension of k.We use G to denote an algebraic group and G(k) to denote the group of k rational points of G. Sometimes we abuse notation and denote the group of Â¯k points of G by G. An algebraic group always means a linear algebraic group unless stated otherwise. The connected component of G is denoted by G0 . The Lie algebra of G is denoted by g. An element g âˆˆ G(k) is k-real if there exists t âˆˆ G(k) such that tgt âˆ’1 = g âˆ’1 . Let H be a subgroup of G. We denote the centralizer of H in G by ZG(H) = {g âˆˆ G | gh = hg âˆ€h âˆˆ H} and the normalizer of H in G by NG(H) = {g âˆˆ G | gHg âˆ’1 = H}. The center of G is denoted by Z(G). The general linear group is denoted by GLn(k) and special linear group by SLn(k). Orthogonal groups are denoted as O(V, b), On(b) or O(q) where b or q indicates the form. Note that we use Sp2n(b) or Sp2n(k) to denote the symplectic group and U(V, h) to denote the unitary group with hermitian form h.The matrix algebra over field k is denoted by Mn(k). We use the symbol C to denote octonion (Cayley) algebras in chapters where we deal with groups of type G2.The symbols âŠ—, âŠ• are used to denote tensor and direct sum respectively. The end of a proof is denoted with symbol Â¤. Bold face word means the word appears for the first time and we give definition for that or a possible reference for the definition.The notation det(A), Hom(M, N), Aut(V ), Gal(K/k) denotes the determinant of a matrix A, the set of all homomorphisms from M to N, the set of all automorphisms of V and the Galois group of field K over k respectively. The symbol diag(A, B, . . . , D) denotes the diagonal matrix where A, B and D themselves are matrices (possibly 1 Ã— 1) sitting on the diagonal. Transpose of a matrix A is written as At and transpose inverse is written as At âˆ’1 .

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